14.4 Derivatives of analytic functions 15.1 Complex series Eugenia Malinnikova, NTNU October 24 2017 1 Derivatives of analytic functions Theorem If f is analytic in some domain D then it has derivatives of any order which are also analytic functions. Moreover, 1 I f (z) f (z0) = dz; 2πi C z − z0 1 I f (z) 0( ) = ; f z0 2 dz 2πi C (z − z0) n! I f (z) (n)( ) = ; f z0 n+1 dz 2πi C (z − z0) where C is a simple closed path in D that bounds some domain in D which contains z0. 2 Examples — I z = π 2 dz 2 i C (z − z0) — I ez 2πiez0 = ; ( z )000 = z 4 dz e e C (z − z0) 6 — I cos z −2πi cos π πi = = ; ( )00 = − 3 dz cos z cos z C (z − π) 24 12 3 Cauchy’s inequality Suppose that f is an analytic function in a disc of radius r around z0 and that jf (z)j ≤ M when jz − z0j = r. Then n!M jf (z)j ≤ M; and jf (n)(z )j ≤ ; for jz − z j ≤ r: 0 r n 0 Let C = fz : jz − z0j = rg, we have n! I f (z) (n)( ) = f z0 n+1 dz 2πi C (z − z0) then taking the absolute values and applying ML-inequality, we get n! M n!M jf (n)(z )j ≤ 2πr = 0 2π r n+1 r n 4 Applications Theorem (Liouville) If a function is analytic in a whole complex plane and bounded in absolute value, then it is a constant Functions analytic in the whole plane are called entire functions. Theorem (Morera) If f is continuous in a simply connected domain D and I f (z)dz = 0 C for any closed curve C. Then f is analytic in D. 5 Series of complex numbers 1 P1 Given a sequence fwngn=1 2 C consider the series n=1 wn The series is convergent if its partial sums have a limit: N X SN = wn ! S; as N ! 1: n=1 This limit is called the sum of the series. Otherwise the series is divergent. Relation to series of real numbers: P Let wn = un + ivn. The series wn converges if and only if each of P P the real series un, vn converges. 6 Basic definitions and facts P P — The series wn is absolutely convergent if jwnj is convergent. P — The series wn is convergent ) wn ! 0 as n ! 1 Some sufficient conditions for convergence PN P — Cauchy criterion: M wn ! 0 as M; N ! 1 , wn converges P — Majorization: an > 0; n = 1; 2;:::, jwnj ≤ an and an P converges ) wn converges; P — Ratio test: jwn+1j=jwnj ≤ q < 1 n = 1; 2;::: ) wn converges; 1=n P — Root test: (jwnj) ≤ q < 1 n = 1; 2;::: ) wn converges 7 The most important example n Geometric series: z 2 C and wn = z 1 ( X 1 ; jzj < 1; zn = 1−z diverges; jzj ≥ 1: 0 Expansion of the Cauchy kernel: Fix z0 2 C and let ζ; z 2 C be such that jz − z0j < jζ − z0j. Then 1 1 1 1 = = z−z = ζ − z (ζ − z0) − (z − z0) ζ − z0 1 − 0 ζ−z0 1 n 1 n 1 X z − z0 X (z − z0) = n+1 ζ − z0 ζ − z0 (ζ − z0) n=0 n=0 We are going to use this formula for making expansions of analytic functions into power series ! 8 Further examples P1 (1+i)n 1. n=1 n! converges (ratio test) P1 (1+i)2n 2n n 2. n=1 2n diverges , j(1 + i) =2 j = 1 6! 0 P1 n+i = P1 1 3. n=1 n2 diverges, the real parts are 1 n and n=1 n diverges n 4. P1 pi converges, n=1 n 1 X in i 1 i 1 p = − p − p + p + ::: n 1 n=1 2 3 4 Separating real and imaginary parts we get two real alternating series, both of them converge and then the series n converge. It does not converge absolutely, P pi diverges. n n 9.